Distributed Control Design for Large-Scale

Interconnected Systems

Cedric Langborta Raffaello D’Andreab

Ramu S. Chandrac

a. Center for the Mathematics of Information, California Institute

of Technology, 1200 E. California Blvd, MS 136-93, Pasadena CA

91125. Email: clangbort@ist.caltech.edu

b. Mechanical & Aerospace Engineering, Cornell University, Ithaca

NY, 14853. Email: rd28@cornell.edu

c. General Electric Research. Email: chandrar@research.ge.com

1

Introduction

In 1974, in his guest editorial for a special issue of the IEEE Transactions on Au-

tomatic Control [1], Richard Bellman emphasized the need for “theories of large

systems”. “This is particularly so”, he added, “since society is composed of large

systems”.

Loosely speaking (we will give a more precise definition later), a large-scale

(interconnected) system is one that is composed of numerous subunits which are

dynamically coupled and/or exchanging information with each other. Examples

abound in our “interconnected age”, where the dependence of our lives on very

large integrated microchips, power grids and communication networks (to name

just a few) has both made Bellman’s statement more obvious and his call for

adapted control tools even more pressing than thirty years ago. As this special

issue illustrates, the field has indeed made much progress to keep up with these

technological developments and has in fact reached a point where it is impossible

to give an exhaustive survey of the literature.

Our goal, in this paper, is to present a set of tools which, although based on

quite recent developments in robust control theory, allow to perform control de-

2

sign of complex networks in the spirit of Bellman’s ideas. In particular, our frame-

work hinges on the concept of convex relaxations and provides a control design

methodology that blends nicely with computation and can thus handle the nu-

merous degrees of freedom, structural constraints, and uncertainties inherent to

large-scale systems.

To emphasize our self-proclaimed affiliation with Bellman’s program, we found

it fitting to structure our paper around some of its quotes and use them as section

headers. In order to keep the presentation accessible to a wide audience, we are

only presenting the main ideas of the theory. Most of the mathematical details and

proofs can be found in [2]-[3]-[4].

“A precise way of handling imprecise systems”

The first step of any control design procedure is modeling. As already pointed out

in Bellman’s paper, models of large-scale interconnected systems are especially

subject to uncertainties. This is not only due to an accumulation of subsystem-

level uncertainties but also to the very nature of the control-oriented modeling

process. Indeed, for the dimension of the full system’s state space to be reason-

3

able and its description to be practical (in the sense that the corresponding data

does not take so much space in a computer’s memory as to render any calculation

hopeless), we typically cannot use the most accurate models of the subsystems

and must rely on reduced order models. The result is uncertainty on the actual

dynamics of the system, which must be taken into account for control design, if

one wants to ensure appropriate behavior on the true system.

Besides, even if we could model every subsystem perfectly, our description

of the full system would still involve some uncertainty, as a result of our poor

knowledge of the coupling mechanisms at play between these units. For example,

in the case of cooperative vehicles transmitting their position to teammates over a

wireless network, we know that the information received by each vehicle will be

degraded (because of quantization and possible delay-inducing congestion in the

network) but the exact characteristics of this degradation depend on many hard-to

evaluate and/or time-varying factors. It is thus safer, for the purposes of control

design, to treat these degradations as uncertainties (by, say, assuming that the true

time-varying transmission delay belongs to an interval) and, again, ensure that the

obtained control law is robust to them.

4

Finally, there is a last type of impreciseness which is both specific and essen-

tial to large-scale interconnected systems. Roughly speaking, it lies in the fact that

different units may know different things about the system, with different degrees

of precision and that, even if they have information regarding the same elements,

their beliefs may be incompatible. This segmentation of information can also be

modeled as uncertainty. For example, if a car on an automated highway only

knows the state of the road at its current position but not at the location where

it is going, and if it believes that the truck in front of it has a functional braking

system, even though the truck has detected a problem with the brakes, it will have

to either acquire this new information or, if impossible, be cautious when deciding

on its next move and hence, use a robust control law.

The framework we propose to handle these three types of uncertainties is that

of robust control theory and Linear Fractional Representations (LFR) [5]. It is

interesting to note that those methods do not use randomness as a model of im-

precision, which is also what Bellman advocated in his text.

5

Modeling

Following the discussion of the previous section, we decide to model large-scale

systems as follows. Consider a network or undirected graph with a dynamical

system sitting at each of its N vertices. The ith such subsystem is represented by

a multi-input/ multi-output (MIMO) mapping Gi satisfying zi

wi

= Gi

di

vi

,

where the vector-valued signals di and zi designate a local disturbance and per-

formance output, respectively. Signals vi and wi correspond to the aggregate cou-

pling signals used by the subsystem, namely

vi =

vi1

...

viN

and wi =

wi1

...

wiN

,

where vij (respectively wij) is the input received by (respectively, output sent to)

subsystem j, with the convention that those vector-valued signals are of dimen-

sion zero if there is no link edge joining subsystems i and j. We will designate

the set of all disturbances and performance outputs by {di} and {zi}, respectively.

More generally, when considering a variable Mi indexed over a set I, the notation

{Mi} will designate the set of all the values of Mi, when index i varies in I, while

6

{Mi}i∈J will be the set of all values of Mi when i varies in a subset J of I.

We refer to Figure 1 for a depiction of such large-scale systems. Later in this

paper, we will focus on linear time-invariant (LTI) subsystems for which the map

Gi admits a state space representation of the form

xi = Aixi + Bivi + Bdi di

wi = Cixi + Divi + Ddi di

zi = Czi xi + Dz

i vi + Dzdi di.

As mentioned earlier, uncertainties can enter the description of the system in sev-

eral ways. For example, the coupling relation between subsystems is typically

poorly known, which can be captured by writing

vij = ∆ijwji (1)

for all pairs of vertices (i, j) joined by an edge. In (1), ∆ij is only known to

belong to a set of input-output operators characterized, for example, by some

upper-bound on their norm. By adding self-loops to the original graph (that is,

edges corresponding to pairs of vertices of the form (i, i)), one can also handle

uncertainties at the subsystem level in this fashion. Regrouping all the uncertain-

ties together on the one hand and all the subsystems’ input/output maps on the

7

Figure 1: A model of large-scale system with N = 5 subsystems. Each subsystem

has a disturbance input di, a performance output zi and shares coupling signals vij

and wij with its neighbors.

8

other hand, we obtain two block-diagonal operators ∆ and G which can be used

to represent the whole system as the feedback interconnection.

Such linear fractional representations (so-called because the resulting input/output

map from {di} to {zi} is a rational function of ∆) have been proposed by several

authors as models of large-scale systems (see, for example, [6]-[7]) and are at the

heart of the design method we are about to present. The main elements of this

method were first introduced in [8].

Stability and Performance

Once we have a model, the next natural question that arises for interconnections

depicted in Figure 1 is that of stability. How can one ensure that, in the absence of

disturbances, the state of every subsystem will go to zero asymptotically in time?

Then, if this is the case, how can we ensure satisfactory disturbance rejection in

the sense that ∑i

∫ ∞

0

‖zi(t)‖2dt < γ∑

i

∫ ∞

0

‖di(t)‖2dt

for some constant γ?

In the context of LFR, one typically answers such questions by resorting to

9

robust stability tools, such as the celebrated (Scaled) Small Gain Theorem [10] or

Integral Quadratic Constraints methods [11]. However, when particularizing them

to large-scale systems, these stability tests should be implementable with no cen-

tralized knowledge of the subsystems’ characteristics. In other words, we want to

be able to infer stability of the full system from the properties of individual sub-

systems, with no single one precisely knowing what its neighbors are. One basic

idea to achieve this goal can be exposed mathematically as follows.

Assume that for each subsystem i, there exists a positive function Vi of state

xi, which vanishes at the origin, and functions Pij of the interconnection variables

vij, wij such that

d

dtVi(xi(t)) <

∑j

Pij(vij(t), wij(t)) for all t (2a)

vij = ∆ijwji for all i, j ⇒∑

i

∑j

Pij(vij(t), wij(t)) = 0, (2b)

then the system is stable, with∑

i Vi as a Lyapunov function. In essence, condi-

tions (2) are a dissipativity result, in the sense of Jan Willems [9]. Each Vi can be

thought of as an energy stored in each system while, for any input vij and output

wij , Pij(vij, wij) corresponds to a (generalized) power received by subsystem i

from subsystem j. Then, condition (2a) means that part of the power received by

10

subsystem i from its environment is degraded while condition (2b) means that the

interconnection conserves power. It is then clear that, if both hold, energy is dis-

sipated overall, driving the system to the origin and thus proving stability. When

dealing with linear time-invariant subsystems as defined before and quadratic sup-

ply rates and storage functions, conditions (2) can be checked by solving a set of

linear matrix inequalities (LMIs), as (2a) is equivalent to

Ai Bi Bdi

I 0 0

Ci Di Ddi

0 I 0

Czi Dz

i Dzdi

0 0 I

t

0 Xi 0 0 0 0

Xi 0 0 0 0 0

0 0 Z11i Z12

i 0 0

0 0 (Z12i )

tZ22

i 0 0

0 0 0 0 I 0

0 0 0 0 0 −I

Ai Bi Bdi

I 0 0

Ci Di Ddi

0 I 0

Czi Dz

i Dzdi

0 0 I

≺ 0

(3)

for all i, where

Z11i := −diag

1≤j≤NXij

Z22i := diag

1≤j≤N(Xji)

Z12i := diag

(−diag

1≤j≤iYij, diag

i<j≤NY t

ji

).

LMIs are particular instances of semi-definite programs (SDPs). The goal is to

minimize a linear function of the matrix variables, subject to the constraint that

11

some other linear function of the variables is a positive-definite matrix. In (3),

the variables are {Xi} and {Xij}, {Yij} and the constraint is in fact written as

a negative-definiteness condition, using the symbol ‘≺’. This formulation is of

course equivalent since a matrix M is positive definite (that is M � 0) if and only

if −M ≺ 0. Also, (3) could be turned into an optimization problem since M ≺ 0

if and only if the smallest t ≥ 0 such that M − tI ≺ 0 is 0.

What makes SDPs particularly attractive, and explains why so much effort

has been devoted recently to characterizing classes of problems in controls and

engineering-at-large that could be re-written in this form (see, for example, [12]),

is the fact that there exist efficient algorithms to approximately solve them in

polynomial time on a computer.

“Approximate policies will play an important role”

The next natural step following system analysis is controller synthesis to guaran-

tee stability and disturbance rejection. For large-scale systems, this process first

requires to choose an architecture that is compatible with the spatially distributed

nature of the problem. Many choices are possible but, for illustration purposes,

12

we will consider the three types of control laws depicted in Figure 2:

(a) a centralized controller, where a single controller receives the measure-

ments of all subsystems and determines all the control actions at once,

(b) a (fully) decentralized controller, where a unit is assigned to each of the

plant’s subsystems, providing it with control action based solely on local

measurements, and

(c) a distributed controller, where these controlling units can share informa-

tion with some of their neighbors over a network.

The main difference between these architectures does not lie so much in the

physical interconnections and/or routing procedures that their implementation re-

quires, as in the various information patterns available to the controllers that is, the

specification of what is known and measured by each controlling unit. For exam-

ple, a centralized controller could in fact be implemented just like the distributed

controller of Figure 2.(c), assuming that the links between controller’s subsystems

transmit all the received measurements in a flooding-like manner. However, dis-

13

(a)

(b)

(c)

Figure 2: Three possible control architectures. (a). A centralized controller, (b). a

(fully) decentralized controller, and (c). a distributed controller.

14

tributed controllers are not centralized because each of the controller’s subsystem

only has a model of the corresponding plant’s subsystem and must thus compute

its inputs using incomplete information.

Therefore, our choice of a particular control law should not only depend on the

perceived amount of information that needs to be exchanged to implement it (the

number of edges in the interconnection graph of the controller, according to which

the decentralized architecture would be most desirable) but also on the possibility

of practically synthesizing it according to its information pattern, as the size of the

whole system increases. From this viewpoint, a centralized controller disqualifies

itself naturally as its design amounts to solving a MIMO control problem with a

large number of inputs, outputs and states; a situation where current optimization-

based methods are known to perform poorly if structure is not taken into account

at the numerical level.

The decentralized and distributed information patterns, on the other hand, can

be tackled for very large systems using the dissipativity conditions exposed in

the previous section. If each of the plant’s subsystem can be rendered dissipative

through interconnection with a controlling subsystem (designed knowing solely

15

this particular subsystem) and the supply rates satisfy condition (2b), then the re-

sulting closed-loop will be stable and reject perturbations appropriately.

Controllers can be computed by solving a synthesis problem for each subsys-

tem, with coupling only occurring through the closed-loop supply rates. When

subsystems are LTI and storage and supply rates are quadratic, decentralized and

distributed control design amount to solving a set of bilinear matrix inequalities

(BMIs). These BMIs result from the application of conditions (3) to a yet un-

known subsystem which depends linearly on the controller’s parameters. As is

well-known in the field of robust control, solving arbitrary BMIs is an NP-hard

problem.

It turns out that the BMIs corresponding to distributed (dynamic output feed-

back) controllers is in fact equivalent to a set of LMIs with additional variables,

while those stemming from decentralized control design do not have this prop-

erty. In loose terms, the reason for this difference is that, in the distributed case,

we have enough degrees of freedom in the closed-loop supply rate matrices and

the signals exchanged between subsystems to remove the non-convexity of the

problem.

16

In summary, what we have seen is that some information patterns give rise

to tractable control design problems while others do not. In fact, strictly speak-

ing, we have only shown the (in)tractability of our method for certain patterns.

However, various other results, such as [14]-[13], suggest that decentralized con-

trol is indeed intractable and, as a result, that distributed architecture should be

preferred.

“Computers will play an important part”

The last ingredient in Bellman’s program was computation. The exact way in

which computers were supposed to enter the picture was unclear but it is already

remarkable that, thirty years ago, he was hinting in the direction that closed-form

analytic solutions (or even the concept of optimality) were probably not the most

appropriate ones for controlling large systems, and that control laws would some-

how have to be designed with (as opposed to merely implemented on) computers.

Interestingly, this vision does not seem to have been universally accepted even

today, as was recalled recently in [15].

17

We have already commented on some of the computation-oriented aspects of

modern robust control methods in general and of our tools in particular. The con-

trol law is obtained as the output of a tractable optimization problem that can be

solved in polynomial time. The desire of turning our design problem into such a

computationally solvable one is what prompted us to use a distributed architecture

for the controller. However, the story does not end here.

Since the models describing large-scale systems are themselves of large di-

mension, a control design algorithm that terminates in polynomial time in this

parameter may still be too slow to be of practical use. One way to speed up the

computations consists in running them in parallel, by assigning a processor to

each of the subsystems, and having it run all the calculations pertaining to that

subsystem. Here is a possible way to perform such a parallelization, based on

decomposition ideas from convex optimization that exploit the natural structure

of the problem. As we explained before, a satisfactory controller with distributed

architecture can be constructed by solving a set of coupled LMIs, which can be

18

re-written in the following form

Find symmetric matrices Xi � 0 and Xij, i, j = 1...N

such that Li(Xi, {Xij}j∈Ni) ≺ 0 for all i

Xij = −Xji.

(4)

In LMI (4), each map Li takes its values in the space of symmetric matrices and,

as before, M ≺ 0 means that matrix M is negative definite. In the case of analy-

sis, variables Xi simply correspond to candidate storage functions while each Xij

corresponds to a candidate supply rate for edge (i, j). For the synthesis LMIs, the

correspondence is not as transparent. In both cases, however, there is one LMI for

each subsystem involving both private (Xi, which is known solely to subsystem i)

and shared ({Xij}) variables, and two LMIs are coupled if and only if the corre-

sponding subsystems can exchange information, that is if and only if j belongs to

the neighborhood Ni of i. In other words, the semi-definite program has inherited

the structure of the plant.

To exploit this structure for parallel computing, we first convert (4) to an opti-

19

mization problem as follows

min{ti},{Xi},{Xij}

∑i

ti

subject to Li(Xi, {Xij}j∈Ni) � tiI, for all i

Xi � 0 ; ti ≥ −1, for all i

Xji = −Xij, for all i, j.

(5)

Problem (4) is feasible if and only if all the optimal t′is for problem (5) are nega-

tive.

Since this new problem is convex in all its variables, we can perform the optimiza-

tion one variable at a time, which means that its optimum is given by

minXij=−Xji

N∑i=1

φi({Xij}j∈Ni)

where each function φi({Xij}j∈Ni) is defined as the optimal value of the following

problem

minXi,ti

ti

subject to Li(Xi, {Xij}j∈Ni) � tiI

Xi � 0 ; ti ≥ −1.

(6)

Hence, all we have to do to solve our original LMI is minimize the function

Φ =∑N

i=1 φi and check the value of each φi at optimum.

20

We cannot perform this optimization using the familiar gradient descent method

because function Φ is not differentiable. Nevertheless, we can build on its con-

vexity and use subgradients instead (see sidebar). Explaining all the details and

properties of subgradient-based optimization algorithms would take use too far

afield and we refer the interested reader to [12]-[16]. For our purposes, we only

need to point out the following facts.

a. For an appropriate choice of positive scalars αk, the update rule

X (k+1)ij = X k

ij + αkgk for all i, j

Φk+1best = min

(Φk

best, Φ({X (k+1)ij })

)converges to the minimum of Φ, where gk is a subgradient of Φ at the point

{X kij}.

b. Subgradient gk can be computed by combining subgradients of functions φi.

c. Subgradients of functions φi are naturally obtained as part of the output of

any primal-dual solver (like the popular SeDuMi) solving problem (6).

This suggests the following simple distributed algorithm to minimize Φ and, hence,

to solve LMI (4):

21

1. Given a value for {X 0ij} and Φ0

best.

2. Compute φi({Xij}j∈Ni) for all i and a corresponding sub-gradient (fact c).

3. All subsystems transmit their subgradient to their neighbors and use the re-

ceived information to compute the relevant components of gk (fact b).

4. All Xij’s are updated using those subgradients (fact a).

5. Iterate until no progress is made.

This algorithm converges to the minimum of Φ while requiring only limited

communication between processors. In practice, however, the subgradients of the

various functions φi’s will not be available at the same time since computing them

involves solving SDP (6), with a different map Li, and hence different conver-

gence time, for each subsystem. As explained in [12]-[17], the basic steps outlined

above can be modified to accommodate such an asynchronous framework. A typ-

ical result is that convergence to the minimum of Φ is still guaranteed as long as

all subgradients are used infinitely often. Overall, the proposed algorithm is thus

22

very well-suited for practical distributed implementation and has been shown to

perform well in cases for systems too large to be handled in a centralized fashion

[16].

Extensions and Connections

The main analysis and synthesis ideas presented above can be adapted to take

advantage of additional structure in large-scale systems, yielding both less con-

servative and even more computationally tractable results.

Consider for example the large-scale system of Figure 3, where all subsystems

are identical and connected only to their nearest neighbors on a ring. Such mod-

els occur naturally in the context of semi-discretized partial differential equations

or regular arrays with periodic boundary conditions. In that case, the large-scale

system is spatially invariant and we can use this symmetry to our advantage.

If we restrict ourselves to checking stability with identical storage functions

23

(a)

(b)

Figure 3: A spatially invariant large scale systems (a) and the corresponding

closed-loop (b). All subunits are identical and interconnected to their neighbors

in an identical manner. Control design is simplified by exploiting this symmetry.

The obtained controller is distributed, with the same interconnection structure as

the plant.24

and supply rates for all systems, that is

Xi = X and Xi(i+1) = −

0 I

I 0

Xi(i−1)

0 I

I 0

for all i,

LMIs (3) collapse to a single one, irrespective of the number of units in the whole

system. The same reduction also occurs for distributed controller synthesis and

allows to design a controller with the same spatial structure as the plant with a

relatively small computational burden.

In fact, an equivalent LMI can also be derived using multidimensional sys-

tem ideas and algebraic transformations as described in [2]. This latter approach

has the additional benefit of addressing spatially invariant systems over general

groups, and always results in a single, computationally attractive, analysis and

synthesis LMI, even in the limit of an infinite number of subsystems. In [4], it

was shown how better closed-loop performance can be guaranteed by using ad-

ditional properties of the group and aggregating subsystems according to central

subgroups. The result is a hierarchical controller, with one unit assigned to every

element of the quotient group.

Another nice feature of those spatially invariant design tools is that they can

25

also be used to tackle some finite extent system with boundary conditions, as

the ones pictured in Figure 4. In this figure, all blocks represent identical sub-

systems except for the end ones which stand for boundary conditions. If some

relation, that we call spatial reversibility, exists between these boundary condi-

tions and the state space representation of the subsystem, it can be shown that the

system of Figure 4 behaves as if embedded in an infinite, spatially invariant one.

This property is analogous to the so-called method of images in partial differential

equations theory, which states that solutions to Laplace’s or the heat equation on

certain bounded domains can be obtained as solutions of the same equation on

an infinite region, once mirror-image singularities are introduced. For example,

consider the heat equation over a finite bar with insulated ends. If we model the

bar as the interval [0, 1] and assume that it is subject to a heat distribution Q(x),

then the temperature profile over [0, 1] is the same as for an infinite bar with heat

distribution

Q(x) =

Q(x) if x ∈ (k, k + 1], k even

−Q(x) if x ∈ (k, k + 1], k odd.

For spatially reversible finite extent systems, a controller can be designed by

applying the method of [2] to the extended spatially invariant system and trun-

26

(a)

(b)

Figure 4: Large-scale system with boundary conditions. (a). If the sub-unit has

some particular properties, this system behaves as if embedded in the periodic

system of Figure 3 and the same, simpler, synthesis methods can be used, (b).

The same reduction can be used for 2-D system as well.

27

cating the obtained control law. Again, the main benefit of this approach is the

simplicity of the obtained LMI. Details on this approach can be found in [18].

An Example

As an illustration of our tools, we present an application to the control of a power

grid with failing transmission lines. The main objectives are to ensure stability

of the grid in the face of such failures, and to give some guarantees on the level

of performance degradation if, in addition, some of the generators experience dis-

turbances. In other words, we are trying to avoid a blackout. Although our pre-

sentation has so far only focused on continuous-time dynamics, we chose to treat

this problem in the discrete-time framework, as a way to demonstrate a possible

extension of the approach. Details can be found in [20].

Following [19], we consider the network of Figure 5, consisting of N inter-

connected load-driving generators. The dynamics of the ith generator is given

by

xi(k + 1) = Rixi(k) + LiIi(k) + Bui ui(k)

Vi(k) = Kixi(k)

(7)

where, at each time k ≥ 0, vector Ii(k) (respectively Vi(k)) contains the real

28

and imaginary part of the current (respectively voltage) deviations from a chosen

operating point, ui(k) is the control torque applied to regulate the generator, and

vector xi(k) is the state of the generator, corresponding to the deviations from the

rotor’s angular velocity’s and angle’s reference profile.

Each generator is connected to a load with admittance matrix Yii and to other

generators through a transmission line of varying admittance ∆ij(k)Yij, j = 1...N .

We assume that the functions ∆ij , which model failure, are taking their values in

{0, 1} but that other characteristics are unknown.

Using Kirchhoff’s laws, we derive that

Ii(k) = YiiVi(k) +∑j 6=i

∆ij(k)Yij(Vi(k)− Vj(k))

and, in turn, that

xi(k+1) = (Ri+LiYiiKi)xi+Li

∑j 6=i

∆ij(k)Yij(Kixi(k)−Kjxj(k))+Bui ui (8)

for all i = 1...N .

Introducing a disturbance di, on the control torque, a measured output yi = xi

and a performance output zti = 1

10

[xt

i uti

], for each generator, equations (8)

can be captured by an LFR as described above, with each subsystem being an LTI

29

Figure 5: (a). A power network with N = 3 generators; (b). The corresponding

model including multiple self-loops.

30

discrete-time system with

Ai := (Ri + LiYiiKi); Bi :=

[Li . . . Li

]︸ ︷︷ ︸

4(N−1)

; Bdi := Bu

i

and coupling signals satisfying

wij = −YijKixi for i 6= j ; wii =

Y1i

...

Y(i−1)i

Y(i+1)i

...

YNi

Kixi.

The interconnection graph has an edge for every pair of connected generators,

along with multiple self-loops adjacent to every vertex (one loop for every neigh-

bor of the corresponding generator).

We applied our synthesis methods to model (8), for the case of N = 10 gen-

erators interconnected over the complete graph and with the following data

Ri =

12

0

0 0

, Li =

1 0

0 0

,

Bui =

−1

0

, Ki =

0 0

0 −1

,

31

Yii =

1 − i10

− i10

1

for all i,

Yij =

1 −1

1 1

for i 6= j.

Our LMIs produced a stabilizing distributed controller with a guaranteed upper-

bound on closed-loop performance of γ = 3.6. Other methods, like [21], assume

a stochastic model for the failures and give necessary and sufficient SDP condi-

tions for the existence of a stabilizing controller with some specified bound on

performance. However, they require adding an LMI for each of the possible fail-

ure modes which, in our case, would involve 245 different cases, assuming that

failures are independent! Our approach, while admittedly more conservative both

in the representation of the failures and on the guaranteed upper-bound, results in

practically tractable computations.

Conclusions and New vistas

With its ability to incorporate inherent uncertainties and its computationally at-

tractive synthesis methods, modern robust control theory has gone a long way in

answering some of Bellman’s questions regarding control of large-scale systems.

32

However, there is one aspect of the problem which has remained mostly un-

touched and, to again use Bellman’s words, constitutes “a good field [...] because

it has not been worked on before”. While we now know how to design controllers

for a large family of networks and respect their underlying, pre-existing topology,

we are still missing the tools for an integrated theory of networks, where topology

and control systems can be designed jointly. Could this be a challenge for the next

thirty years?

A Side Bar: Subgradients

Subgradients generalize the concept of gradient for non-differentiable, convex

functions.

Let f : U → R be a real-valued convex function defined over an open convex

subset of Rn. A vector v ∈ Rn is called a subgradient of f at x if, for any y ∈ U

f(y)− f(x) ≥ v.(y − x),

where ‘.’ denotes the usual scalar product. The set of all subgradients of f at x is

called the subdifferential at x.

33

In words, the subdifferential at x corresponds to the set of hyperplanes which,

passing through (x, f(x)), leave the graph of f entirely in one of the two half-

spaces they define. For example, if f is the ‘absolute value’ function x 7→ |x|,

the subdifferential at 0 is the interval [−1, 1], while the only subgradient at x > 0

(respectively x < 0) is 1 (respectively −1).

It can be shown that, for convex functions, the subdifferential is always a non-

empty, convex, compact set at every point. Also f has a local minimum at a point

x if and only if the zero vector belongs to the subdifferential at x.

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